What is Moment of inertia: Definition and 1000 Discussions
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.
It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis.
For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.
A stick of uniform density with mass M = 7.5 kg and length L = 1 m is pivoted about an axle which is perpendicular to its length and located 0.12 m from one end. Ignore any friction between the stick and the axle.
a) What is the moment of inertia of the stick about this axle?
Iaxle = kg...
Hi,
An old GRE problem asks what the moment of inertia of seven pennies, arranged in a hexagon with one in the center, all touching each others' edges is, about the axis that passes through the center of the central penny and is normal to the plane of the pennies.
Each penny is a uniform...
Homework Statement
Derive an expression for the moment of inertia about the axis of symmetry for
a cylinder of mass M , length L and radius a, where the mass density decreases as a
function of distance from the axis as 1/r
Homework Equations
The Attempt at a Solution
1) am i...
Homework Statement
For a little background, this lab was on energy of a rolling object. We rolled a hollow cylinder from the top of a ramp on a table and onto the floor. We are trying to experimentally derive the constant (k) found in the equation for moment of inertia.
Variables: h =...
Homework Statement
2 children, masses 35kg and 40kg sit at opposite ends of a 3.4m seesaw with mass 25kg with the fulcrum at midpoint. with the seesaw horizontal find
a) the net torque
b)angular acceleration
Homework Equations
tau=rF=Ialpha
The Attempt at a Solution
i got the right...
Help: Moment of inertia Ixx, Iyy, Izz, Ixy, Ixz, Iyz, etc??
Hi all,
Can someone help with a few equations?, i need to know the moment of inertia of a section.
The section is a column which in the Z direction have 3m, on the X direction has 0.3m and on the Y direction has 0.6m.
I know...
Homework Statement
see attachment
Homework Equations
The Attempt at a Solution
Hey can someone please tell me what I've done wrong here.
There is all so a question further down that asks for the same thing but with mass of the bar included.(M_bar)
Homework Statement
Consider a particle of mass = 19.0 revolving around an axis with angular speed . The perpendicular distance from the particle to the axis is = 0.500 .
Homework Equations
I=mr^2
K=0.5Iw^2
The Attempt at a Solution
I=4.75
K=1796 which is 0.5*4.75*27.5^2...
Moment of Inertia - Unsolved
Homework Statement
A 22 kg solid door is 220 cm tall, 92 cm wide.
a) What is the door's moment of inertia for rotation on its hinges?
b) What is the door's moment of inertia for rotation about a vertical axis inside the door, 17 cm from one edge?
Homework...
Homework Statement
A block with mass m = 5.00 kg slides down a surface inclined 36.9 to the horizontal. The coefficient of kinetic friction is 0.25. A string attached to the block is wrapped around a flywheel has mass 25.0 kg and moment of inertia 0.500 kgm2 with respect to the axis of...
Homework Statement
A block with mass m = 5.00 kg slides down a surface inclined 36.9 to the horizontal. The coefficient of kinetic friction is 0.25. A string attached to the block is wrapped around a flywheel has mass 25.0 kg and moment of inertia 0.500 kgm2 with respect to the axis of...
Homework Statement
A 15.0 kg bucket of water is suspended by a very light rope wrapped around a solid cylinder 0.300 m in diameter with a mass of 12.0 kg. The cylinder pivots on a frictionless axle through its centre. The bucket is released from rest at the top of a well and falls 10.0 m to the...
Homework Statement
A 15.0 kg bucket of water is suspended by a very light rope wrapped around a solid cylinder 0.300 m in diameter with a mass of 12.0 kg. The cylinder pivots on a frictionless axle through its centre. The bucket is released from rest at the top of a well and falls 10.0 m to...
Homework Statement
The pulley in the diagram has a radius of 0.160 m and a moment of inertia 0.480 kg.m². The rope does not slip on the pulley rim. Use energy methods to calculate the speed of the 4.00 kg block just before it strikes the floor
Homework Equations
The Attempt...
Homework Statement
Consider a hollow tube of mass M = 1.2 kg and length L = 1.6 m that rotates about an axle through its center and perpendicular to its length. Inside the tube are two masses, m_1 = 0.4 kg each. These masses are initially held a distance d = 0.8 m apart by a string and...
Homework Statement
A holiday ornament in the shape of a hollow sphere with mass 1.0×10−2 kg and radius 5.0×10−2 m is hung from a tree limb by a small loop of wire attached to the surface of the sphere. If the ornament is displaced a small distance and released, it swings back and forth as a...
Homework Statement
Ok i have the following problem. There is a pulley with radius 7.5 cm and 2 forces acting on it (2 tensions of a string) acting in opposite directions either side. One is 18.2N, the other 7.5N. I know the acceleration is 3.75 meteres per second squared.
Find the moment...
Homework Statement
I am working on a lab i did. In the lab we tracked the motion of two pucks that collided. One puck collided into another stationary puck of equal mass in an inelastic collision. We used a video capture system to record the velocity of the pucks before and after...
Homework Statement
A side view of a car tire. Model it as having two sidewall of uniform thickness 0.635 cm and a tread wall of uniform thickness 2.50 cm and width 20.0 cm. Assume the rubber has a uniform density equal to 1.10 X 10^3 kg/(m^3). Radius from the center of the tire to the rubber...
Homework Statement
A mass (M) is dropped from height (H) onto one end of a stick of mass (M) and of length (L) pivoted around the opposite end. Upon collision the mass adheres to the stick. Respond to the following in terms of M, L, H, and g.
a. Find speed of mass just before impact
b. Find...
Hello. I am attempting to derive the equation for the moment of inertia of a filled sphere. I do not how how to do the proper notation by typing, so I have attached a scan of my work(I apologize if opening the attachment is inconvenient)
I keep getting the correct order final equation, but my...
Homework Statement
2) Three spherical masses are located in a plane at the positions shown in the figure below. A
has mass 39.3 kg, B has mass 35.9 kg, and C has mass 17.6 kg.
Calculate the moment of inertia (of the three masses) with respect to the z-axis perpendicular to the xy plane...
Assuming that I have a C-Channel
When the load acts along the X-X axis, we take Iyy for the Bending moment calculations. Similarly, when load acts along the Y-Y axis, we take Ixx. But if i have a load that acts on Z-Z axis, how do I calculate the Moment of Inertia?
PLS SEE THE ATTACHMENTS...
The problem statement
A stiff uniform wire of mass M0 and length L0 is cut, bent, and the parts soldered together so that it forms a circular wheel having four identical spokes coming out from the center. None of the wire is wasted, and you can neglect the mass of the solder. What is the moment...
This was a 2 part problem...
PART A: calculate moment of inertia of a uniform sphere of mass M and radius R by using the information provided:
the moment of inertia of a thin spherical shell at radius R with mass m spinning about its axis is 2/3mR2.
I did this by integrating over thin...
Homework Statement
A uniform disc radius a, mass m
there is rotation about an axis (z) tangental to the disc and in the plane of the disc.
a point mass m is placed at the centre of the disc.
what is the new moment about the axis z
also show that the period of oscillations will...
Homework Statement
Use spherical coordinates to find the moment of inertia about the z-axis of a solid of uniform density bounded by the hemisphere \rho=cos\varphi, \pi/4\leq\varphi\leq\pi/2, and the cone \varphi=4.
Homework Equations
I_{z} = \int\int\int(x^{2}+y^{2})\rho(x, y, z) dV...
Homework Statement
calculate the moment of inertia of a sphere of mass M and radius R by integrating over thin shells
Homework Equations
Ishell=(2/3)mR2The Attempt at a Solution
this is what i have so far
the sphere is decomposed into infinitesimal shells with surface area 4\pir2
the mass of...
Homework Statement
The four masses shown in the attachment are connectec by massless, rigid rods.
a) find the coordinates of the center of mass.
b) find the moment of inertia about a diagonal axis that passes through mass B and D.
Homework Equations
I=MR^2...I think.
The Attempt...
Homework Statement
Homework Equations
Moment of Inertia equation: I=(1/12)bh^3
Stress and Strain equations: Sigma= -(M*c)/I Tao = (V*Q)/(I*t)
The Attempt at a Solution
I know I need to use shear and moment diagrams to find the max moment so I can use the Stress diagram to...
Homework Statement
A 1.65 kg mass is attached to a light cord that is wrapped around a pulley of radius 4.65 cm, which turns with negligible friction. The mass falls at a constant acceleration of 2.40 m/s2. Find the moment of inertia of the pulley.
Homework Equations
I=mr^2
Torque=I*alpha...
Homework Statement
Determine the moment of inertia of a cylinder
of radius 0.37 m, height 1.8 m and density
(1.11 − 0.555 r + 0.284 r2) kg/m3 about the
center.
Answer in units of kg · m2.
Homework Equations
I=Integral[(r^2)dm]
I = mr^2
Density = Mass/Volume
Volume of a cylinder =...
Homework Statement
Consider a 4 kg square which has its mass concentrated along its perimeter, with each side of length 6 m.
(a) What is the moment of inertia of the square about an axis perpendicular to the plane of the square at its center of mass? Use the parallel axis theorem and divide...
Homework Statement
A uniform rectangular coil of total mass 270 g and dimensions 1m x 5m is oriented perpendicular to a uniform 4.00-T magnetic field (the figure ). A current of 2.90 A is suddenly started in the coil.
Homework Equations
angular acceleration = torgue / moment of inertia...
1. Homework Statement .
A very thin 1.0 kg disk with diameter 80 cm is mounted horizontally to rotate freely about a central vertical axis. On the edge of the disk, sticking out a little, is a small, essentially mass-less tab, or "catcher." A 1.0 g wad of clay is fired at a speed of 10.0 m/s...
I seem to be off by a factor of 2 on the answer to this problem but I can't find where I went wrong. The term in front should be 1/5 and not 2/5. Does anybody see the mistake in my work? It is attached in a word document because I can't figure out how to put the equations into this post...
Ok, so I thought about a derivation for the moment of inertia, but my answer comes out to (3/5)MR^2
Basically, what I did was I considered the sphere as a sum of infinitesimally thin spherical shells.
The moment of inertia for one shell is dI=(r^2)*dm
where dm=(M/V)*4*pi*r^2*dr
where...
Ok, so I thought about a derivation for the moment of inertia, but my answer comes out to (3/5)MR^2
Basically, what I did was I considered the sphere as a sum of infinitesimally thin spherical shells.
The moment of inertia for one shell is dI=(r^2)*dm
where dm=(M/V)*4*pi*r^2*dr
where...
Homework Statement
A thin rectangular slab, with dimensions 0.580 m by 0.830 m and mass 0.150 kg, is rotated about an axis passing through the slab parallel to the short edge. If the axis is 0.230 m from the short edge, what is the moment of inertia of the slab?
Homework Equations...
Homework Statement
A triangular prism of mass M, whose two ends are equilateral triangles parallel to the xy plane with side 2a, is centered on the origin with its axis on the z axis. Find its moment of inertia for rotation about the z axis. Without doing any integrals write down and explain...
Homework Statement
A vehicle wheel has a mass 2.5 kg and diameter 310 mm. The vehicle moves from rest to a linear velocity of 13 kmh -1 in 60 s. Assuming that the wheel can be treated as a solid disc, find the torque applied to the wheelHomework Equations
Not so sure on how to add...
Homework Statement
Consider a rigid body with an inertia tensor I =[30, 0, 0; 0, 40, 0; 0, 0, 20] N m s^2 and angular velocity w=10j+10k rad/s. Determine the moment of inertia about an axis parallel to w and find the rotational kinetic energy.
The attempt at a solution
I'm not sure...
Homework Statement
There are 2 models of rugbeaters. Model A has a 1m long handle and a 40cm edge length square. The handles mass is 1kg and the squares mass is 0.5kg. Model B has a 0.75m long handle and a square that has 30cm edge length. The mass of the handle is 1.5kg and the mass of the...
Hlw...I am a new guy here...And I am taking a course of Applied Mechanics in my freshmen year...Here I have come across a term called Area of moment of inertia which I don't really understand that much...And I have been given to prove that
I_d = \frac{\pi a^4}{4}
...I was...
Hi Guys,
I've written a clutch model for my simulation based off a few papers I've read which basically deal with it as a state machine; that is there are two separate equations to integrate the motion when it is either locked or slipping. I'm interested to find out if this can be dealt with...
Homework Statement
This was a TMSCA (Texas Math and Science Competition) from the physics section.
A thin meter stick of mass 200g is positioned vertically on a frictionless table. It is released, slips, and falls. Find the speed of its center of mass just before it hits the table.Homework...
Homework Statement
In this experiment a physical pendulum consisting of a rod and two large rubber stoppers (on top of each other) is swung from a point 1 mm from the end point. Calculate the moment of inertia of the pendulum about the point of suspension?
Data:
m=mass of rod=97.1 g...
I am confused about one thing on this derivation. Okay so the guide I am following goes like this..
-------------------------------------
\sigma=\frac{M}{A}
dm=\sigma dA=(\frac{M}{4\pi R^2})2\pi rsin\phi Rd\phi
dm=\frac{M}{2}sin\phi d\phi
This is one part that confuses me. It seems as...
Homework Statement
I have a drum that comprises of basically a cylinder and rotated line in the form y=mx+c
http://img22.imageshack.us/img22/6108/druma.jpg
Homework Equations
I=\int r^2 dm
The Attempt at a Solution
I found the MOI for the cylinder as 1/2M(a2+b2) where a and b...